def test_derivative(self):
"""Check that numerical and analytical derivatives of Q match."""
spde = SphereMeshSPDE(level=3)
Q = spde.build_Q_stationary(numpy.log(1.0), numpy.log(1.0), 2)
epsilon = 0.000001
expected_dQ0 = \
((spde.build_Q_stationary(numpy.log(1.0)+epsilon, numpy.log(1.0), 2) -
spde.build_Q_stationary(numpy.log(1.0)-epsilon, numpy.log(1.0), 2)) / (2.0 * epsilon)).todense()
expected_dQ1 = \
((spde.build_Q_stationary(numpy.log(1.0), numpy.log(1.0)+epsilon, 2) -
spde.build_Q_stationary(numpy.log(1.0), numpy.log(1.0)-epsilon, 2)) / (2.0 * epsilon)).todense()
dQ0 = spde.build_dQdp_stationary(numpy.log(1.0), numpy.log(1.0), 2,
0).todense()
dQ1 = spde.build_dQdp_stationary(numpy.log(1.0), numpy.log(1.0), 2,
1).todense()
# print numpy.abs(dQ0 - expected_dQ0).ravel().max() / numpy.abs(expected_dQ0).ravel().max()
# print numpy.abs(dQ1 - expected_dQ1).ravel().max() / numpy.abs(expected_dQ1).ravel().max()
numpy.testing.assert_almost_equal(dQ0, expected_dQ0, decimal=7)
numpy.testing.assert_almost_equal(dQ1, expected_dQ1, decimal=7)
def test_covariance_function(self):
"""Check that inverse precision follows matern covariance patterns."""
# Generate
spde = SphereMeshSPDE(level=5)
# Index 12 in the mesh is 0 latitude, 0 longitude
# - The covariance evaluated will relate to distance from this chosen point
# but the same shape of graph should be obtained for any point -
# there is nothing special about this choice.
reference_index = 12
# nu and rho as used in matern definition
nu = 1.0
rho = 0.3
# Make precision
precision = spde.build_Q_stationary(log_sigma=numpy.log(
numpy.sqrt(1.0)),
log_rho=numpy.log(2.0 * rho),
alpha=2)
# Sample covariance at reference index
# This is the vector with 1 in the reference location and zeros everywhere else
# like [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ... ]
reference_unit = scipy.sparse.csc_matrix(
([1.0], ([reference_index], [0])), shape=(precision.shape[0], 1))
# Solve Q.x = reference unit
# This is like picking the column of inverse(Q) corresponding to reference index
# and so is the covariance with respect to that reference location
covariance_wrt_reference = scipy.sparse.linalg.spsolve(
precision, reference_unit)
# Great arc distances from reference point
points = spde.triangulation.points
refpoint = points[reference_index, :]
dotprod = points.dot(refpoint)
distance_from_reference = numpy.arccos(dotprod)
# There is a scale factor difference between estimated covariance and Matern model
sigma0 = numpy.sqrt(covariance_wrt_reference[reference_index])
# Compute disparity (taking into account scale factor difference)
disparity = max(
numpy.abs(covariance_wrt_reference - TestSphereMeshSPDE.matern(
distance_from_reference, sigma0, nu, rho)))
# Convert to percentage and check
disparity_percent = 100.0 * disparity / (sigma0**2)
# print 'Disparity (%): ', disparity_percent
self.assertTrue(disparity_percent < 2.0)